Optimal. Leaf size=4 \[ -\cot (x) \]
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Rubi [A] time = 0.0135559, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3175, 3767, 8} \[ -\cot (x) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{1-\cos ^2(x)} \, dx &=\int \csc ^2(x) \, dx\\ &=-\operatorname{Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)\\ \end{align*}
Mathematica [A] time = 0.0022625, size = 4, normalized size = 1. \[ -\cot (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 7, normalized size = 1.8 \begin{align*} - \left ( \tan \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956744, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59319, size = 22, normalized size = 5.5 \begin{align*} -\frac{\cos \left (x\right )}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.596599, size = 14, normalized size = 3.5 \begin{align*} \frac{\tan{\left (\frac{x}{2} \right )}}{2} - \frac{1}{2 \tan{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17903, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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